Related papers: On the asymptotic linearity of reduction number

We prove here that the elements of any standard basis of $I^n$, where $I$ is an ideal of a Noetherian local ring and $n$ is a positive integer, have order bounded by a linear function in $n$. We deduce from this that the elements of any…

We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider…

We call a graded connected algebra $R$ effectively coherent, if for every linear equation over $R$ with homogeneous coefficients of degrees at most $d$, the degrees of generators of its module of solutions are bounded by some function…

Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group…

Let $(R,\fr m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M, N$ two finitely generated $R$-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that…

Results are presented concerning the following question: If M is a finitely-generated module with finite local cohomologies over a Noetherian local ring (A, m), does there exist an integer n such that every parameter ideal for M contained…

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

Given a commutative Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix…

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring $R$ with infinite residue field and positive depth. In this paper, we answer a question of…

The purpose of the present paper is to prove some properties of the strongly irreducible submodules in the arithmetical and Noetherian modules over a commutative ring. The relationship among the families of strongly irreducible submodules,…

Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty.$ Rees proved that for all $n\gg0$, $\lambda(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$…

We define simplicial and dimension $\Gamma$-groups, the generalizations of simplicial and dimension groups to the case when these groups have an action of an arbitrary group $\Gamma.$ Assuming that the integral group ring of $\Gamma$ is…

An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv…

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a…

It is well known that in the Noetherian local ring with infinite residue field the reduction of $\mm$-primary ideal may be given in the form of a sufficiently general linear combination of its generators. In the paper we give a condition…

The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behaviour of the minimum distance of projective Reed-Muller…

Let $\Lambda$ be the von Mangoldt function and $(r_G(n) = \sum_{m_1 + m_2 = n} \Lambda(m_1) \Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N \geq 2$ be an integer. We prove that $$\begin{align} &\sum_{n \le N} r_G(n)…

Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study…

The main goal of this paper is to compare the silting theory of an $R$-algebra $\Lambda$ over a Noetherian ring $R$ with that of its tensor product $\Lambda \otimes \Gamma$ with another $R$-algebra $\Gamma$. In the case that the $R$-algebra…